Conjugate and cut time in the sub-Riemannian problem on the group of motions of a plane

Sachkov, Yu. L.

doi:10.1051/cocv/2009031

Cited by 97 publications

(81 citation statements)

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“…We also computed the exact lower and upper bound of the n-th conjugate time. We discovered an unexpected symmetry in the Jacobian expression and the conjugate points in the case of oscillating and rotating pendulum which hasn't been observed in optimality analysis in sub-Riemannian problem on SE(2) [18], the Engel group [19] and the Euler elastic problem [17]. We conclude that the n-th conjugate time is bounded by similar functions from below and above for both λ ∈ C 1 and λ ∈ C 2 .…”

Section: Resultsmentioning

confidence: 68%

“…A lower bound of the form t conj 1 (λ) ≥ t M AX 1 (λ) for all extremal trajectories q(t) = Exp(λ, t) was proved in the Euler Elastic problem [17], sub-Riemannian problem on SE(2) [18] and sub-Riemannian problem on the Engel group [19] via homotopy considering the fact that the Maslov index (number of conjugate points along an extremal trajectory) is invariant under homotopy [20]. In order to qualify for proof of absence of conjugate points below the lower bound of the first conjugate time via homotopy, the optimal control problem must satisfy a set of hypotheses (H1)-(H4) [17] outlined below.…”

Section: Conjugate Points and Homotopymentioning

confidence: 99%

“…2. Along each normal extremal trajectory, conjugate times are isolated one from another, see [17], [18].…”

Section: Conjugate Points and Homotopymentioning

confidence: 99%

“…We employ the approach adopted in the proof of Theorems 2.1, 2.2 [18] and prove that for λ ∈ C 1 the interval (0, 2K(k)) does not contain conjugate points for the extremal trajectory q(t) = Exp(λ, t).…”

Section: Theorem 41 the First Conjugate Time Formentioning

confidence: 99%

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Maxwell Strata and Conjugate Points in the Sub-Riemannian Problem on the Lie Group SH(2)

2016

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We study local and global optimality of geodesics in the left invariant sub-Riemannian problem on the Lie group SH(2). We obtain the complete description of the Maxwell points corresponding to the discrete symmetries of the vertical subsystem of the Hamiltonian system. An effective upper bound on the cut time is obtained in terms of the first Maxwell times. We study the local optimality of extremal trajectories and prove the lower and upper bounds on the first conjugate times. We also obtain the generic time interval for the nth conjugate time which is important in the study of sub-Riemannian wavefront. Based on our results of n-th conjugate time and n-th Maxwell time, we prove a generalization of Rolle's theorem that between any two consecutive Maxwell points, there is exactly one conjugate point along any geodesic.

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Section: Resultsmentioning

confidence: 68%

Section: Conjugate Points and Homotopymentioning

confidence: 99%

“…2. Along each normal extremal trajectory, conjugate times are isolated one from another, see [17], [18].…”

Section: Conjugate Points and Homotopymentioning

confidence: 99%

Section: Theorem 41 the First Conjugate Time Formentioning

confidence: 99%

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Maxwell Strata and Conjugate Points in the Sub-Riemannian Problem on the Lie Group SH(2)

2016

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“…In a forthcoming paper [21] we prove that in fact t cut = t. The bound on the cut time is obtained via the study of discrete symmetries of the problem and the corresponding Maxwell points -points where two distinct sub-Riemannian geodesics of the same length intersect one another.…”

Section: Introductionmentioning

confidence: 97%

Maxwell strata in sub-Riemannian problem on the group of motions of a plane

Моисеев¹,

Sachkov²

2009

ESAIM: COCV

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Abstract.The left-invariant sub-Riemannian problem on the group of motions of a plane is considered. Sub-Riemannian geodesics are parameterized by Jacobi's functions. Discrete symmetries of the problem generated by reflections of pendulum are described. The corresponding Maxwell points are characterized, on this basis an upper bound on the cut time is obtained. Mathematics Subject IntroductionProblems of sub-Riemannian geometry have been actively studied by geometric control methods, see books [2,3,7,10]. One of the central and hard questions in this domain is a description of cut and conjugate loci. Detailed results on the local structure of conjugate and cut loci were obtained in the 3-dimensional contact case [1,6]. Global results are restricted to symmetric low-dimensional cases, primarily for left-invariant problems on Lie groups (the Heisenberg group [5,22], the growth vector (n, n(n + 1)/2) [9,11,12], the groups SO(3), SU(2), SL(2) and the Lens Spaces [4]).The paper continues this direction of research: we start to study the left-invariant sub-Riemannian problem on the group of motions of a plane SE(2). This problem has important applications in robotics [8] and vision [13]. On the other hand, this is the simplest sub-Riemannian problem where the conjugate and cut loci differ one from another in the neighborhood of the initial point.The main result of the work is an upper bound on the cut time t cut given in Theorem 5.4: we show that for any sub-Riemannian geodesic on SE(2) there holds the estimate t cut ≤ t, where t is a certain function defined on the cotangent space at the identity. In a forthcoming paper [21] we prove that in fact t cut = t. The bound on the cut time is obtained via the study of discrete symmetries of the problem and the corresponding Maxwell points -points where two distinct sub-Riemannian geodesics of the same length intersect one another.This work has the following structure. In Section 1 we state the problem and discuss existence of solutions. In Section 2 we apply Pontryagin Maximum Principle to the problem. The Hamiltonian system for normal extremals is triangular, and the vertical subsystem is the equation of mathematical pendulum. In Section 3Keywords and phrases. Optimal control, sub-Riemannian geometry, differential-geometric methods, left-invariant problem, Lie group, Pontryagin Maximum Principle, symmetries, exponential mapping, Maxwell stratum. * The second author is partially supported by Russian Foundation for Basic Research, and we endow the cotangent space at the identity with special elliptic coordinates induced by the flow of the pendulum, and integrate the normal Hamiltonian system in these coordinates. Sub-Riemannian geodesics are parameterized by Jacobi's functions. In Section 4 we construct a discrete group of symmetries of the exponential mapping by continuation of reflections in the phase cylinder of the pendulum. In the main Section 5 we obtain an explicit description of Maxwell strata corresponding to the group of discrete symmetries, and prove the upper ...

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